This document contains mathematical formulas and definitions related to limits, derivatives, and integrals. Some key points: 1) It defines limits as x approaches a or infinity, including one-sided limits and limits at infinity. Functions and expressions are taken to limits. 2) Rules for limits are presented, such as limits of sums, products, quotients. Continuity and differentiability are defined in terms of limits. 3) The derivative is defined as the limit of the difference quotient. Properties of derivatives are listed, including derivatives of polynomials, trig functions, and composite functions. 4) Definite integrals are defined using limits of Riemann sums. Properties of integrals include integrals of sums, products

1. ªW
ª
ğ­
ª
ª
ª
:

2. 2

3. 3
@ @
WW
Jğ4
ªJġ5
ª6
8
¯10
JJú12
13
14
Ğn16
ª 18
20
22
24
26
28
ª 31
32
34
ª36
ª ğ38

4. 4
ğK
ğWax b+( )0a ≠
+∞
b
a
−−∞x
aaax b+
ğW²ax bx c+ +( )0a ≠
W( ) ²x ax bx cΡ = + +
WWWW
( ) 0x x∈ Ρ =»
( )xΡ( )xΡ
0∆
S = ∅
+∞−∞x
a( )xΡ
Ù
Ħ
0∆ ={ }2
b
S
a
−
=
+∞
b
a
−−∞x
aa
( )xΡ
( )
²
2
b
x a x
a
 Ρ = +   
= b² - 4ac∆
0∆
{ };1 2S x x=
W
1 2
b
x
a
− − ∆
=
2 2
b
x
a
− + ∆
=
+∞2x1x
−∞
x
a
a
a
( )xΡ
F­ØW1 2x xE
( ) ( )( )1 2x a x x x xΡ = − −
1x2xW( )0 ² 0a x ax bx c≠ ∈ + + =»
W1 2
b
x x
a
−
+ =1 2
c
x x
a
× =

5. 5
ª
ġK
ªW
Ħab
( )2 2 2
2a b a ab b+ = + +
( )2 2 2
2a b a ab b− = − +
( )( )2 2
a b a b a b− = − +
( )3 3 2 2 3
3 3a b a a b ab b+ = + + +
( )3 3 2 2 3
3 3a b a a b ab b− −− = +
( )( )3 3 2 2
a b a b a ab b− = − + +
( )( )3 3 2 2
a b a b a ab b−+ = + +
ġW
PQĦ
fÙ Ù Ù Ù xĨĨĨĨWWWWġġġġfWWWW
( ) ( )f x x= ΡfD = »
( )
( )
( )
x
f x
Q x
Ρ
=( ){ }/ 0fD x Q x= ∈ ≠»
( ) ( )f x x= Ρ( ){ }/ 0fD x x= ∈ Ρ ≥»
( )
( )
( )
x
f x
Q x
Ρ
=( ){ }/ 0fD x Q x= ∈ »
( )
( )
( )
x
f x
Q x
Ρ
=( ) }0Q x ( ){ / 0fD x x= ∈ Ρ ≥»
( )
( )
( )
x
f x
Q x
Ρ
=( ) }0Q x ≠
( )
( )
/ 0f
x
D x
Q x
 Ρ= ∈ ≥

»

6. 6
ª)Ù(K
ª( )* nn x x∈ »x xW
0
lim 0n
x
x
→
=0
lim 0
x
x
→
=
1
lim 0n
x x→−∞
=
1
lim 0n
x x→+∞
=
lim
x
x
→+∞
= +∞
1
lim 0
x x→+∞
=
nWWWWnWWWW
lim n
x
x
→+∞
= +∞
lim n
x
x
→−∞
= +∞
0
1
lim n
x x→
= +∞
0
1
lim n
x x→
= +∞
lim n
x
x
→+∞
= +∞
lim n
x
x
→−∞
= −∞
0
1
lim n
x x→
= +∞
0
1
lim n
x x→
= −∞
Ğ ğ ª+∞−∞W
+∞−∞
×
+∞−∞
×
ªW
0
sin
lim 1
x
x
x→
=
0
tan
lim 1
x
x
x→
=
0
1 cos 1
lim
² 2x
x
x→
−
=
ªW( )x u x
( )
0
lim
x x
u x
→
( )
0
lim
x x
u x
→
0≥ll
+∞+∞
ğ ª0xĦ0x+∞−∞

7. 7
Ø ªW
( ) ( )
( )
( )
0
0
lim
lim 0 x x
x x
f x V x
f x
V x →
→
− ≤  ⇒ == 
l
l
( ) ( ) ( )
( )
( )
( )
0 0
0
lim lim
lim
x x x x
x x
u x f x V x
u x f x
V x
→ →
→
≤ ≤ = ⇒ =
= 
l l
l
( ) ( )
( )
( )
0
0
lim
lim x x
x x
u x f x
f x
u x →
→
≤  ⇒ = +∞= +∞
( ) ( )
( )
( )
0
0
lim
lim x x
x x
u x V x
f x
V x →
→
≤  ⇒ = −∞= −∞
ğ ª0xĦ0x+∞−∞
ª ªW
Ħ ġW
+∞+∞−∞lll
( )
0
lim
x x
f x
→
−∞+∞−∞+∞−∞'l
( )
0
lim
x x
g x
→
+∞−∞+∞−∞l + l'( ) ( )[ ]
0
lim
x x
g x f x
→
+
ĦW
0+∞−∞−∞0l0ll
( )
0
lim
x x
f x
→
±∞+∞+∞−∞+∞−∞+∞−∞'l
( )
0
lim
x x
g x
→
+∞−∞+∞+∞−∞−∞+∞×l l'( ) ( )[ ]
0
lim
x x
g x f x
→
×
ĦW
±∞0+∞−∞0l0lll
( )
0
lim
x x
f x
→
±∞00+
0−
0+
0−
0+
0−
0+
0−±∞0≠'l( )
0
lim
x x
g x
→
+∞−∞−∞+∞+∞−∞−∞+∞0
'
l
l
( )
( )0
lim
x x
g x
f x→
W
ğ ª0xĦ0x+∞−∞

8. 8
K
W
W( ) ( )
0
0lim
x x
f f x f x
→
⇔ =0x
Ħ–W
•( ) ( )
0
0lim
x x
f f x f x
→
⇔ =Ħ0x
•( ) ( )
0
0lim
x x
f f x f x
→
⇔ =0x
fĦ0f x⇔0x
ġW
fª ġ] [,a bfÓ] [,a b
fġ[ ],a bfª Ó] [,a b
Ħab
ªW
fgġ Ħ ĦIk
•f g+f g×kfÓI
•gIĦ
1
g
f
g
Ó ĦI
W•»
•ġ
•x x+
»
•sinx xcosx x»
•tanx xġ{ }/
2
k k
π
π− + ∈» »
ĦW
fġIgġJęW( )f I J⊂
Wg fοÓI
ġW
•
•ġ ġ
ªWfġI
ĞÓ( )f I

9. 9
ÓÓÓÓ( )f I
ÓÓÓÓI
fIfI
[ ],a b( ) ( )[ ];f a f b( ) ( )[ ];f b f a
[ [,a b( ) ( ); lim
x b
f a f x−→
 
 
  
( ) ( )lim ;
x b
f x f a−→
 
 
  
] ],a b( ) ( )lim ;
x a
f x f b
+→
 
 
  
( ) ( ); lim
x a
f b f x
+→
 
 
  
] [,a b( ) ( )lim ; lim
x bx a
f x f x−+ →→
 
 
  
( ) ( )lim ; lim
x b x a
f x f x− +→ →
 
 
  
[ [,a +∞( ) ( ); lim
x
f a f x
→+∞
 
 
  
( ) ( )lim ;
x
f x f a
→+∞
 
 
  
] [,a +∞( ) ( )lim ; lim
xx a
f x f x
+ →+∞→
 
 
  
( ) ( )lim ; lim
x x a
f x f x
+→+∞ →
 
 
  
] ],a−∞( ) ( )lim ;
x
f x f a
→−∞
 
 
 
( ) ( ); lim
x
f a f x
→−∞
 
 
 
] [,a−∞( ) ( )lim ; lim
x x a
f x f x−→−∞ →
 
 
  
( ) ( )lim ; lim
xx a
f x f x− →−∞→
 
 
  
»( ) ( )lim ; lim
x x
f x f x
→−∞ →+∞
 
 
  
( ) ( )lim ; lim
x x
f x f x
→+∞ →−∞
 
 
  
×W
fġ[ ],a bβĦ ( )f a( )f b
αÓ[ ],a bęW( )f α β=
Wfġ[ ],a b( ) ( ) 0f a f b×
( ) 0f x =αÓ ħ[ ],a b
fġ[ ],a b( ) ( ) 0f a f b×
( ) 0f x =αÓ ħ[ ],a b
W
fġ[ ],a bęW( ) ( ) 0f a f b×
αğ( ) 0f x =Ó [ ],a b
WWWW( ) 0
2
a b
f a f
 +
 ×    
WWWW( ) 0
2
a b
f b f
 +
 ×    
W
2
a b
a α
+
Ù
2
b a−
Ó ī;
2
a b
a
 +
 
  
¯ Ùα
W
2
a b
bα
+
Ù
2
b a−
Ó ī;
2
a b
b
 +
 
  
¯ Ùα
WÙ ğ ħ īαª

10. 10
¯K
¯W
f ¯0xW
( ) ( )0
0 0
lim
x x
f x f x
x x→
−
−
f0xW( )0'f x
J W
f ¯0x
f 0xW( )( ) ( )0 0 0'y f x x x f x= − +
u»W( ) ( )( ) ( )0 0 0'u x f x x x f x= − +
f 0xfĘ0x
Ħ ¯JĦ ¯W
f Ħ ¯0x
( ) ( )
0
0
0
lim
x x
f x f x
x x→
−
−
fĦ0xW( )0'f xd
f¯0xW
( ) ( )
0
0
0
lim
x x
f x f x
x x→
−
−
f0xW( )0'f xg
f ¯0xf Ħ ¯0x( ) ( )0 0' 'f x f xg d=
¯W
f ¯0xf0x
ªW
( )f x′( )f x
0k( )k ∈ »
1x
1
²x
−1
x
1rrx −rx{ }( )* 1r ∈ −»
1
2 x
x
cosxsinx
sinx−cosx
2
2
1
1 tan
cos
x
x
+ =tanx

11. 11
ª-ĦJĞW
( )u v u v′ ′ ′+ = +( )u v u v′ ′′− = −( ) ( ) ( )k ku k u′ ′∈ =»
( )uv u v uv′ ′′ = +( ) 1
.n n
u nu u −′ ′=
( )1
²
v
v v
′ ′−
=( ) ²
u u v uv
v v
′ ′ ′−
=
( )u v u v vο ο ′ ′′ = × ( )
2
u
u
u
′
′ =
ª Ù ¯W
fġ ¯I
( ) 0f x I f x′⇔ ∀ ∈ ≥ÓI
( )' 0f x I f x⇔ ∀ ∈ ≤ÓI
( )' 0f x I f x⇔ ∀ ∈ =ÓI
¯W
( )fCWWWW
( ) ( )
( )0
0
0
lim
0x x
f x f x
a
x x a→
−
=
− ≠
( )( )0 0;A x f xa
( ) ( )
0
0
0
lim 0
x x
f x f x
x x→
−
=
−
f ¯
0x
( )( )0 0;A x f x
( ) ( )
( )
0
00
lim
0
f x f x
a
x xx x a+
−
=
−→ ≠
Ħ( )( )0 0;A x f x
a
( ) ( )0
00
lim 0
f x f x
x xx x +
−
=
−→
f¯
Ħī0x
Ħ( )( )0 0;A x f x
( ) ( )0
00
lim
f x f x
x xx x +
−
= −∞
−→
Ħ
( )( )0 0;A x f x
( ) ( )0
00
lim
f x f x
x xx x +
−
= +∞
−→
fÙ
¯
Ħī0x
Ħ
( )( )0 0;A x f x
( ) ( )
( )
0
0 0
lim
0
f x f x
a
x x x x a
−
−
=
→ − ≠
( )( )0 0;A x f x
a
( ) ( )0
0 0
lim 0
f x f x
x x x x−
−
=
→ −
f¯
0x
( )( )0 0;A x f x
( ) ( )0
0 0
lim
f x f x
x x x x−
−
= −∞
→ −
( )( )0 0;A x f x
( ) ( )0
0 0
lim
f x f x
x x x x−
−
= +∞
→ −
fÙ
¯
0x
( )( )0 0;A x f x

12. 12
–
úK
W
x a=ĩ ( )fC
ĜW
•( )2f fx D a x D∀ ∈ − ∈
•( ) ( )2fx D f a x f x∀ ∈ − =
W
( ),I a bĩ( )fC
ĜW
•( )2f fx D a x D∀ ∈ − ∈
•( ) ( )2 2fx D f a x f x b∀ ∈ − + =
–ª-úW
ġ
Ó Ĝ
W( ) 0x I f x′′∀ ∈ ≤
W( )fCÓI
ġ
¯Ó
W( ) 0x I f x′′∀ ∈ ≥
W( )fCÓI
ú
Ù
f ′′0xÙ
( )fC ú0x
f ′0xÙ
( )fC ú0x

13. 13
K
( ) ( )[ ]lim 0
x
f x ax b
→∞
− + = ( )
( )0
lim
x a
f x
a
x→∞ ≠
=
( )lim
x
f x
→∞
= ∞
( )
lim
x
f x
x→∞
= ∞
( )
lim 0
x
f x
x→∞
=
( )[ ]lim
x
f x ax b
→∞
− = ( )[ ]lim
x
f x ax
→∞
− = ∞
( )fCW
W
x a=
( )fCW
ě
Ę∞
( )fCW
ě
Ę∞
( )fCW
ě
y ax=
Ę∞
( )fCW
W
y ax b= +
Ę∞
( )fCW
W
y a=
Ę∞
( )lim
x a
f x
→
= ∞( )lim
x
f x a
→∞
=

14. 14
K
WfġI
fÓ( )f IÓ I
W1
f −
W
•
( ) ( )
( )
1
f x y f y x
x I y f I
− =  =  ⇔ 
 ∈ ∈  
•( )( )
1
x I f f x xο−
∀ ∈ =
•( ) ( )( )1
y f I f f y yο −
∀ ∈ =
ĜW
fġI
xÓ( )f IyÓI
W( ) ( )1
f x y f y x−
= ⇔ =
yx( )
1
f x−
x( )f I
W
fġI
1
f −
Ó( )f I
¯W
fġI
0xÓ( )f I( )0 0y f x=
f ¯0x( )0' 0f x ≠
1
f −
¯0y
W( ) ( )
( )
1
0
0
' 1
'
f y
f x
−
=
fġI
fÓ ¯If ′ÓI
1
f −
Ó ¯( )f I
W( ) ( ) ( )
( )
1
1
' 1
'
x f I f x
f f x
−
−
∀ ∈ =
 
  

15. 15
W
fġI
1
f −
Ùf
W
ªW
( )fC( )1fC −
( ) ( ), fA a b C∈( ) ( )1' ,
f
A b a C −∈
Wx a=Wy a=
Wy b=Wx b=
Wy ax b= +
W
1 b
y x
a a
= +
ĜW
x ay b= +
FEFE
FEFE
fġI
Ħ f1
f −

16. 16
Ğ( )*n n∈ »
ĞK
W
Wn
x x+
»Ğn
Wn
n
nx x
+
→» »++++
::::
( ) 2
; nnx y x y x y+∀ ∈ = ⇔ =»
ªW
•2x x=
•W3 x Ğx
ªW
( )
( )
2
; *
nn
nn
n n
n n
x y n
x x
x x
x y x y
x y x y
+∀ ∈ ∀ ∈
=
=
= ⇔ =
⇔
» »( ) ( ) ( )
( )
( )
22 *
; ;
0
n nn
m mnn
n
n
n
n m n m
x y m n
x y x y
x x
x x
y
y y
x x
+
×
∀ ∈ ∀ ∈
× = ×
=
= ≠
=
» »
W
x y
x y
x y
−
− =
+
3 3
3 3 33² ²
x y
x y
x x y y
−
− =
+ +
ġW
fWWWWġġġġfWWWW
( ) nf x x=[ [0;fD = +∞
( ) ( )nf x u x=( ) }0u x ≥{ /f uD x x D= ∈ ∈»
ªW
ğ ª0xĦ0x+∞−∞
( )
0
lim
x x
u x
→
( )
0
lim n
x x
u x
→
0≥ln
l
+∞+∞

17. 17
W
nx x+
»
uġI
uġI( )nx u xÓI
¯W
nx xÓ ¯] [0;+∞
W
] [ ( )
1
1
0; n
n n
x x
n x −
′∀ ∈ +∞ =
uġI
uġ ¯I
( )nx u xÓ ¯I
W( )( )
( )
( )[ ] 1
n
nn
u x
x I u x
n u x −
′′∀ ∈ =
W( ) n
a x x a∈ ∈ =» »
nn
0a { }nS a={ };n nS a a= −
0a ={ }0S ={ }0S =
0a { }nS a= −S = ∅
ĞW
p
r
q
=ÙW*
p ∈ »*
q ∈ »
] [0,
p
q qr p
x x x x∀ ∈ +∞ = =
ªW
•] [
1
0; n nx x x∀ ∈ +∞ =
•ġfÙ xW( ) ( ) ( )[ ]* r
r f x u x∈ =»
W( ) }0u x { /f uD x x D= ∈ ∈»
•( )( ) ( )( ) ( ) ( )[ ]
1 1
11
'n
n nu x u x u x u x
n
−
′ ′ = = × ×  
xy*
+»rr′*»
•( )
' 'rr r r
x x ×
=•
' 'r r r r
x x x +
× =
•
r r
r
x x
y y
  =  
•( )r r r
x y x y× = ×
•
'
'
1 r
r
x
x
−
=•
r
r r
r
x
x
x
′−
′
  =   

18. 18
ª K
ğ – W
1n nu u r+ = +
r
1n nu q u+ = ×
q
ğğğğ
( )n pu u n p r= + −
( )p n≤
n p
n pu u q −
×=
( )p n≤
ġġġġ1
1
...
1
n p
p n p
q
u u u
q
− + −  + + = ×   − 
1
1
...
1
n p
p n p
q
u u u
q
− + −  + + = ×   − 
( )1q ≠
abc
2b a c= +²b a c= ×
–:
( )n n I
u ∈
•( ) nn n I
u n I u M∈
⇔ ∀ ∈ ≤M
•( ) nn n I
u n I u m∈
⇔ ∀ ∈ ≥m
•( )n n I
u ∈( )n n I
u ∈
⇔
W
( )n n I
u ∈
•( ) 1n nn n I
u n I u u+∈
⇔ ∀ ∈ ≤
•( ) 1n nn n I
u n I u u+∈
⇔ ∀ ∈ ≥
•( ) 1n nn n I
u n I u u=+∈
⇔ ∀ ∈

19. 19
W
( )nα
W*α ∈ »W
0α 0α
lim
n
nα
→+∞
= +∞lim 0
n
nα
→+∞
=
( )n
qWq ∈ »W
1q 1q =1 1q− 1q ≤ −
lim n
n
q
→+∞
= +∞lim 1n
n
q
→+∞
=lim 0n
n
q
→+∞
=
( )n
q
ªW
•
•
lim lim
lim
n n n
n n
n n
n
n
v u w
v u
v
→+∞ →∞
→+∞
≤ ≤ = ⇒ =
= 
l l
l
lim
lim 0
n n
n
nn
n
u v
u
v →∞
→+∞
− ≤  ⇒ =
= 
l
l
lim
lim
n n
n
nn
n
u v
u
v →+∞
→+∞
≤  ⇒ = −∞= −∞
lim
lim
n n
n
nn
n
u v
u
v →+∞
→+∞
≥  ⇒ = +∞= +∞
( )1nu f un+ =W
×( )nuW
( )
0
1n n
u a
u f u+
=

=
fġIę( )f I I⊂aI
( )nulW( )f x x=

20. 20
K
ġW
W
fġI
FfÓI
ĜW
•FÓ ¯I
•( ) ( )'x I F x f x∀ ∈ =
ªW
Ó ġ
fġI
FfÓIW
fIĨW
( ) ( )x F x k k+ ∈ »
fġI
0xI0y»
FfÓI
ĜW( )0 0F x y=
WĦ Ó-ĞW
W
fgĦġ Ħ ĦIk
FGĦ Ħ ĦfgÓIW
•F G+f g+ÓI
•kFkfÓI

21. 21
W
( )F x( )f x
ax k+a ∈ »
1
²
2
x k+x
1
k
x
−
+
1
²x
2 x k+
1
x
1
1
r
x
k
r
+
+
+
r
x{ }( )* -1r ∈ −»
cosx k− +sinx
sinx k+cosx
tanx k+
1
1 tan²
cos²
x
x
+ =
ln x k+
1
x
( )k ∈ »x
ke +x
e
¯W
( )F x( )f x
( )2 u x k+
( )
( )
'u x
u x
( )
1
k
v x
+
( )
( )[ ]
'
²
v x
v x
−
( )[ ] 1
1
r
u x
k
r
+
+
+
( ) ( )[ ]' r
u x u x×{ }( )* -1r ∈ −»
( )ln u x k+
( )
( )
'u x
u x
( )u x
ke +( )
( )
' u x
u x e×
( )
1
sin ax b k
a
+ +( )cos ax b+( )0a ≠
( )k ∈ »( )
1
cos ax b k
a
− + +( )sin ax b+( )0a ≠

22. 22
ª ğK
W
WfġIFfÓI
abÓI
faħbğW
( ) ( )[ ] ( ) ( )
b
a
b
f x dx F x F b F a
a
= = −∫
ªW
ĠW
( ) 0
a
f x dx
a
=∫( ) ( )
a b
f x dx f x dx
b a
= −∫ ∫
( ) ( ) ( )
b b
k kf x dx k f x dx
a a
∈ =∫ ∫»( ) ( )[ ] ( ) ( )
b b b
f x g x dx f x dx g x dx
a a a
+ = +∫ ∫ ∫
W
( ) ( ) ( )
b c b
f x dx f x dx f x dx
a a c
= +∫ ∫ ∫
ØW
W[ ] ( ), 0x a b f x∀ ∈ ≥
W( ) 0
b
f x dx
a
≥∫
W[ ] ( ) ( ),x a b f x g x∀ ∈ ≤
W( ) ( )
b b
f x dx g x dx
a a
≤∫ ∫
W
fġ[ ],a b
Ó ğW( )
1 b
f x dx
ab a− ∫
W
uvġ ¯ Ħ ĦIĦ ęu′v′Ó ĦI
abÓI
( ) ( ) ( ) ( )[ ] ( ) ( )b
a
b b
u x v x dx u x v x u x v x dx
a a
′ ′= −∫ ∫
ħ ( ), ,o i j
.u AÒ oĦ ij
1. .u A i j= ×

23. 23
fġ[ ],a b
Ħ Ò ğfC
Ħ W
x a=x b=
W( ) . .
b
f x dx u A
a
    ∫
fgġ Ħ Ħ[ ],a b
Ħ Ħ Ò ğfCgC
Ħ W
x a=x b=W
W( ) ( ) . .
b
f x g x dx u A
a
  −   ∫
ªW
ªªªªğğğğWWWW
f
Ó[ ],a b
( ) . .
b
f x dx u A
a
    ∫
f
Ó[ ],a b
( ) . .
b
f x dx u A
a
  −   ∫
•f
Ó[ ],a c
•f
Ó[ ],c b
( ) ( ) . .
c b
f x dx f x dx u A
a c
  + −   ∫ ∫
( )fC¯( )gC
Ó[ ],a b
( ) ( )( ) . .
b
f x g x dx u A
a
  −   ∫
•( )fC¯( )gC
Ó[ ],a c
•( )gC¯( )fC
Ó[ ],c b
( ) ( )( ) ( ) ( )( ) . .
c b
f x g x dx g x f x dx u A
a c
  − + −   ∫ ∫
ª:
Ó( )fC
ġ [ ];a b
W( )( )² .
b
V f x dx u v
a
π
 
 =
  
∫
uvWğ

24. 24
K
Ù
W
Ù
1
x
x
Ó] [0; +∞
1Wln
ª ªW
ln 1e =ln1 0=
] [ ] [0; 0;x y∀ ∈ +∞ ∀ ∈ +∞
ln lnx y x y= ⇔ =
ln lnx y x y ⇔
] [0;
ln y
x y
x y x e
∀ ∈ +∞ ∀ ∈
= ⇔ =
»
] [ ] [
( )
( )
0; 0;
ln ln ln
ln ln
1
ln ln
ln ln ln
r
x y
xy x y
x r x
x
x
x
x y
y
∀ ∈ +∞ ∀ ∈ +∞
= +
=
  = −  
  = −  
( )r ∈ »
nW( )* ln lnn
x x n x∀ ∈ =»
ġW
fWWWWġġġġfWWWW
( ) ( )[ ]lnf x u x=( ) }0u x ‫و‬{ /f uD x x D= ∈ ∈»
( ) ( )( )2
lnf x u x=  
 
( ) ( )lnf x u x=
( ) }0u x ≠‫و‬{ /f uD x x D= ∈ ∈»
ªW
( )lim ln
x
x
→+∞
= +∞ln
lim 0n
x
x
x→+∞
=
( )
0
lim ln
x
x
→
= −∞
( )
0
lim ln 0n
x
x x
→
=
1
ln
lim 1
1x
x
x→
=
−
( )
0
ln 1
lim 1
x
x
x→
+
=
( )n *∈»
W
lnx xÓ] [0;+∞
uġI
uġI( )[ ]lnx u xÓI

25. 25
¯W
lnx x¯] [0;+∞
:
] [ ( )
1
0; lnx x
x
′∀ ∈ +∞ =
uġI
uġ ¯I
W( )[ ]lnx u xÓ ¯I
W( )[ ]( )
( )
( )
''
ln
u x
x I u x
u x
∀ ∈ =
WlnW
+∞10x
+-lnx
aW{ }*
1a +
∈ −»
Wa Waogl
W] [ ( )
ln
0;
ln
a
x
x og x
a
∀ ∈ +∞ =l
ª ªW
1 0
1
a
aa
og
og
=
=
l
l
] [ ] [0; 0;
og og
og
a a
r
a
x y r
x y x y
x r x a
∀ ∈ +∞ ∀ ∈ +∞ ∀ ∈
= ⇔ =
= ⇔ =
»
l l
l
] [ ] [
( )
( )
0; 0;
1
a a a
r
a a
a a
a a a
x y
og xy og x og y
og x r og x
og og x
x
x
og og x og y
y
∀ ∈ +∞ ∀ ∈ +∞
= +
=
  = −  
  = −  
l l l
l l
l l
l l l
( )r ∈ »
ªª:
1a 0 1a
a aog x og y x y ⇔ l la aog x og y x y ⇔ l l
0
lim
lim
a
x
a
x
og x
og x+
→+∞
→
= +∞
= −∞
l
l
0
lim
lim
a
x
a
x
og x
og x+
→+∞
→
= −∞
= +∞
l
l
W
] [ ( )
1
0, '
ln
ax og x
x a
∀ ∈ +∞ =l

26. 26
K
Ù
W
ÙÙ
Wexp
x»( )exp x
x e=
ª ªW
0x
x e∀ ∈ »
( )ln x
x e x∀ ∈ » =
] [ ln
0, x
x e x∀ ∈ +∞ =
] [0;
lnx
x y
e y x y
∀ ∈ ∀ ∈ +∞
= ⇔ =
»
( ); ² x y
x y
x y e e x y
e e x y
∀ ∈ = ⇔ =
⇔
»
x y∀ ∈ ∀ ∈» »
x y x y
e e e +
× =
( )r ∈ » ( )
rx rx
e e=
1 x
x e
e
−
=
x
x y
y
e
e
e
−
=
ġW
fWWWWġġġġfWWWW
( )
x
f x e=fD = »
( )
( )u x
f x e={ }/f uD x x D= ∈ ∈»
ªW
lim x
x
e
→+∞
= +∞
lim 0x
x
e
→−∞
=
lim
x
n
x
e
x→+∞
  = +∞  
( )lim 0n x
x
x e
→−∞
=
0
1
lim 1
x
x
e
x→
−
=
( )n *∈»
W
x
x e»
uġI
uÓI
( )u x
x eÓI

27. 27
¯W
x
x e¯»W( )x x
x e e′∀ ∈ =»
uġI
u¯ÓIW( )u x
x eÓ ¯I
W( )
( ) ( )
( )'u x u x
x I ue x e′∀ ∈ = ×
lnW
aW}{1a ∗
+∈ −»
WaoglaWexpa
x»( )exp x
a x a=
ªªW
lnx x a
x a e∀ ∈ =»
( )x
aog a x=l
] [ ( )
0; og xax a a∀ ∈ +∞ =l
( ) 2
; x y
x y a a x y∀ ∈ = ⇔ =»
] [0;x y∀ ∈ ∀ ∈ +∞»
( )aog ylx
a y x= ⇔ =
( ) 2
;x y∀ ∈ »
x y x y
a a a +
× =
( )r ∈ » ( )
rx rx
a a=
1 x
x a
a
−
=
x
x y
y
a
a
a
−
=
ªªW
1a 0 1a
x y
a a x y⇔ x y
a a x y⇔
lim x
x
a
→+∞
= +∞
lim 0x
x
a
→−∞
=
lim 0x
x
a
→+∞
=
lim x
x
a
→−∞
= +∞
0
1
lim ln
x
x
a
a
x→
−
=
W
( ) ( )lnx x
a a a′ = ×

28. 28
K
ġW}² 1i = −( ){ / ; ²z a ib a b= = + ∈» »
×ĞW
z a ib= +W( ); ²a b ∈ »
•a ib+×Ğz
•ağ ĞzW( )Re z
•bĞzW( )Im z
W•W( )Im 0z =z
•W( )Re 0z =( )Im 0z ≠zĝ
ĦW
zz′Ħ
( ) ( )Im Imz z′=( ) ( )Re Rez z z z′ ′= ⇔ =
W
ħ ( )1 2, ,o e e
W
z a ib= +W( ); ²a b ∈ »
zWz a ib= −
( )M z( )M z′ğ
•' 'z z z z+ = +
•' 'z z z z× = ×
•n n
z z=( )*n ∈ »
•
1 1
' 'z z
  =  
•
' '
z z
z z
  =  
( )' 0z ≠
•z z z⇔ =
•z z z⇔ = −ú ĝ
•( )2Rez z z+ =
•( )2 Imz z i z− =
•( )[ ] ( )[ ]² ²Re Imzz z z= +
W
z a ib= +W( ); ²a b ∈ »
z( ),M a b
•zğMMzW( )M z
•z ğOMW( )OM z( )z Aff OM=
z a ib= +W( ); ²a b ∈ »
z ğW² ²z zz a b= = +

29. 29
( )
( )
*
' 0
n nz z n
z z
z z
z
z z
= ∈
− =
= ≠
′ ′
»
1 1
z z z z
z z
z z
′ ′× = ×
=
=
′ ′
ÙW
zÙM
zθ ªW( ),1 OMe
Wargz
W[ ]arg 2z θ π=
zÙ
r z=[ ]arg 2z θ π=
•zW
( ) [ ]cos sin ,z r i rθ θ θ= + =
•zWi
z re θ
=
ªW
aÙ
0a 0a
[ ],0a a=
,
2
ai a
π
= +
 
 
  
[ ],a a π= −
,
2
ai a
π
= − −
 
 
  
•( ) ( )[ ]arg ' arg arg ' 2zz z z π≡ +
•[ ]arg arg 2z z π≡ −
•( )[ ]arg arg 2z zπ π− ≡ +
•[ ]arg arg 2n
z n z π≡
•[ ]
1
arg arg 2z
z
π≡ −−−−
•( ) [ ]arg arg arg ' 2
'
z
z z
z
π≡ −
•[ ] [ ] [ ], ', ' '; 'r r rrθ θ θ θ× = +
•[ ] [ ], ,r rθ θ= −
•[ ] [ ], ,r rθ π θ− = +
•[ ], ;n n
r r nθ θ =  
•
[ ]
1 1
; '
'; ' 'r r
θ
θ
 
= − 
  
•
[ ]
[ ]
;
; '
'; ' '
r r
r r
θ
θ θ
θ
 
= − 
  
•( )''
' ' ii i
re r e rr e θ θθ θ +
× =
•i i
re reθ θ−
=
•( )ii
re re π θθ +
− =
•( )
ni n in
re r eθ θ
=
•'
'
1 1
''
i
i
e
rr e
θ
θ
−
=
•( )'
' ''
i
i
i
re r
e
rr e
θ
θ θ
θ
−
=
[ ] [ ], 2 ,k r k rθ π θ∀ ∈ + =»
•argz z kπ⇔ =‫د‬
•arg
2
z z k
π
π⇔ = +‫د‬‫ف‬( )k ∈ »
WÙW
( ) ( ) ( )cos sin cos sinn
n
i n n i nθ θ θ θ
∀ ∈
+ = +
»( )1
cos
2
i i
e eθ θ
θ θ −
∀ ∈ = +»
( )1
sin
2
i i
e e
i
θ θ
θ −
= −
²z z a∈ =»( )a ∈ »W
WWWWġġġġWWWW
0a { };S a a= −
0a ={ }0S = ²z z a∈ =»
0a { };S i a i a= − − −

30. 30
W² 0z az bz c∈ + + =»Wabc( )0a ≠
WWWW ġ ġ ġ ġWWWW
0∆ ;
2 2
b b
S
a a
  − − ∆ − + ∆ =  
   
0∆ ={ }2
b
S
a
−
=
( )
2
2
0
4
z az bz c
b ac
∈ + + =
∆ = −
»
0∆ ;
2 2
b i b i
S
a a
  − − −∆ − + −∆ =  
   
ªW
ABB AAB z z= −
I[ ];A B
2
A B
I
z z
z
+
=
( );AB AC( ) [ ]; arg 2c A
B A
z z
AB AC
z z
π
 −  ≡    − 
ABC
C A
B A
z z
z z
−
∈
−
»
ABCD
D A B C
B A D C
z z z z
z z z z
− −
× ∈
− −
»D A D C
B A B C
z z z z
z z z z
− −
× ∈
− −
»
Az z r− =
( )0r
•AM r=
•M ħAr
A Bz z z z− = −
•AM BM=
•Mħ[ ]AB
;
2
C A
B A
z z
r
z z
π−  
= ± 
 −  
ABCA
[ ]1;C A
B A
z z
z z
θ
−
=
−
ABC ĦA
1;
2
C A
B A
z z
z z
π−  
= ± 
 −  
ABC ĦA
1;
3
C A
B A
z z
z z
π−  
= ± 
 −  
ABC
ª ĩªW
ĩĩĩĩWWWW
t ªuz z b′ = +b ğu
hk( )z k zω ω′ − = −ωğ
rθ( )
i
z e zθ
ω ω′ − = −ωğ

31. 31
ªK
WWWWğğğğWWWW
'y ay b= +
( )0a ≠
( )
ax b
y x e
a
α= −
( )α ∈ »
WWWWWWWW WWWWğğğğWWWW
0∆
Ħ Ħ
Ħ 1r2r
( ) 1 2r x r x
ey x e βα +=
W( ), ²α β ∈ »
0∆ =r
( ) ( ) rx
ey x x βα +=
W( ), ²α β ∈ »
'' ' 0y ay by+ + =
( )
² 0
² 4
r ar b
a b
+ + =
∆ = −
0∆
Ħ Ø Ħ ĦW
1r p iq= −
2r p iq= +
( ) ( )cos sin px
y x qx qx eα β= +
W( ), ²α β ∈ »

32. 32
K
ħ ¯ ( ), , ,o i j k
WĞJJ Ğ
( ), ,u a b c( )', ', 'v a b cĦ3ϑ
•. ' ' 'u v aa bb cc= + +
•² ² ²u a b c= + +
•
'
' ' '
'
' ' '
'
i a a
b b a a a a
u v j b b i j k
c c c c b b
k c c
∧ = = − +
W
Ħ ĦABW
( ) ( ) ( )² ² ²B A B A B AAB x x y y z z= − + − + −
M( )P0ax by cz d+ + + =W
( )( ),
² ² ²
M M Max by cz d
d M
a b c
+ + +
Ρ =
+ +
M( ),A u∆W( )( ),
AM u
d A
u
∧
∆ =
W
( ) ( ), , : 0n a b c ax by cz d⇔ Ρ + + + =( )P
ABCÙAB AC∧( )ABC
Ĝ ī( )ABCW
( ) ( ). 0M ABC AM AB AC∈ ⇔ ∧ =
W
( ), ,a b cRW
( ) ( ) ( )² ² ² ²x a y b z c R− + − + − =

33. 33
( )S[ ]ABĜ ī
W( ) . 0M S AM BM∈ ⇔ =
W( )S[ ]AB
2
AB
( ),S R( ): 0ax by cz dΡ + + + =
H( )Ρ
W( )( );d H d= = Ρ
( )P( )S( )P( )S
H
( )P( )S
( )C
WH
W
2 2
r R d= −
( ),S R( )∆W
H( )∆
W( )( );d H d= = ∆
( )∆( )S( )∆( )S
H
( )∆( )S
Ħ Ħ

34. 34
K
ġW
W
ġEÓEWCardE
W0Card∅ =
W
ABġ
( ) ( )Card A B CardA CardB Card A B∪ = + − ∩
ġW
W
AġE
AE ÓWA
{ }/A x E x A= ∈ ∉
ªW
•A A∩ = ∅
•A A E∪ =
•cardA cardE cardA= −
W
ě ×p( )*p ∈ »
ª1n
ª2n
.........................................
pªpn
Ğ W1 2 3 ... pn n n n× × × ×
ª Ø-ª ØW
ª ØW
np*»( )p n≤
تpĦnWp
n

35. 35
ª ØW
np*»( )p n≤
تpĦnW
( ) ( ) ( )1 2 ... 1p
nA n n n n p= × − × − × × − +
p
W
nĦnn
W( ) ( )! 1 2 ... 2 1n n n n= × − × − × × ×
ªW
Eġn
AEp( )p n≤
pĦn
ªW
!
p
p n
n
A
C
p
=
W!np
nAp
nC
( ) ( )! 1 2 ... 2 1
0! 1
n n n n n∗
∈ = × − × − × × ×
=
»
( )
!
! !
p
n
n
C
p n p
=
−( )
!
!
p
n
n
A
n p
=
−
1n
nC =1
nC n=0
1nC =1n
nC n−
=
p n p
n nC C −
=1
1
p p p
n n nC C C−
++ =
W
pĦn( )p n≤
Ğ W
WWWW ª ª ª ªWWWWØØØØ
p
nCÙ
p
n
p
nA

36. 36
ªK
ª
WWWWWWWW
ěě
ª ª ġ
AAª
ğ ĜA B∩ğ ĜAB
ğ ĜA B∪ĜAB
ğAğA( )A A A A∩ = ∅ ∪ =‫و‬
ABĦ ÙA B∩ = ∅
JW
Wě ª
•{ }iωipğ{ }iωWip
W{ }( )i iP pω =
•ğ ª ġ
{ }1 2 3; ; ;...; nA ω ω ω ω=ğAW
( ) ( ) ( ) ( ) ( )1 2 3 ... np A p p p pω ω ω ω= + + + +
ªWě ª
•( ) 0p ∅ =( ) 1p =
•( )0 1p A≤ ≤A
•Ħ ĜĦ ĜĦ ĜĦ ĜWWWW
ĦAB
( ) ( ) ( ) ( )p A B p A p B p A B∪ = + − ∩
( ) ( ) ( )p A B p A p B∪ = +ABĦ Ù
• ğ ğ ğ ğWWWW
AW( ) ( )1p A p A= −
ªW
Wě
AW( )
cardA
p A
card
=

37. 37
JĦW
WABę Ħ ĦW( ) 0p A ≠
BğAW( ) ( ) ( )
( )
p A BBp B p A p AA
∩
= =
WĦABę ĦW( ) ( ) 0p A p B× ≠
W( ) ( ) ( ) ( ) ( )B Ap A B p A p p B p BA∩ = × = ×
WĦABĦ
( ) ( ) ( )A p A B p A p B⇔ ∩ = ×B
Wě ª12ě
)1 2∪ =( 1 2∩ = ∅
AW( ) ( ) ( ) ( ) ( )1 2
1 2
A Ap A p p p p= × + ×
ªW
Aě p
ªnğ Ĝk A,W
( ) ( ) ( )1k n kk
nk n C p p −
≤ −
ÙW
Ùě ª
Ù XĦ Ħ W
•Ĝ( ) { }1 2 3; ; ;...; nX x x x x=WÙ ġX
•( )ip X x=iÓ{ }1;2;...;n
JJÙ ú W
nx...3x2x1xix XÙ
Ğ úWnp...3p2p1p( )ip X x=
ÙÙÙÙXWWWW( ) 1 1 2 2 3 3 ... n nE X x p x p x p x p= × + × + × + + ×
Ù Ù Ù Ù XWWWW( ) ( ) ( )[ ]² ²V X E X E X= −
W
Ù ú Ù ú Ù ú Ù ú XWWWW( ) ( )X V Xσ =
ğW
pAě {n
Ù Xª ğ Anp
{ } ( ) ( )0;1;2;...; 1 n kk k
nk n p X k C p p −
∀ ∈ = = × × −
( )E X n p= ×( ) ( )1V X np p= −

38. 38
ª ğ)Ù(K
W
Ħ ªW
-1 cos 1
-1 sin 1
cos² sin² 1
x
x
x x
≤ ≤
≤ ≤
+ =
sin
tan
cos
1
1 tan²
cos²
x
x
x
x
x
=
+ =
( )
( )
( )
cos 2 cos
sin 2 sin
tan tan
x k x
x k x
x k x
π
π
π
+ =
+ =
+ =
ªW
- 2x a kπ= +cos cos 2x a x a kπ= ⇔ = +
( )- 2x a kπ π= +sin sin 2x a x a kπ= ⇔ = +
( )tan tanx a x a k kπ= ⇔ = + ∈ »
2
π
3
π
4
π
6
π
0x
1
3
2
2
2
1
2
0sinx
0
1
2
2
2
3
2
1cosx
31
3
3
0tanx
2
x
π
+
2
x
π
−+xπxπ −x−
cosxcosx-sinxsinx-sinxsin
sinx−sinx-cosx-cosxcosxcos

39. 39
ġ ĜW
( )
( )
( )
cos cos cos - sin sin
sin sin cos cos sin
tan tan
tan
1 - tan tan
a b a b a b
a b a b a b
a b
a b
a b
+ = × ×
+ = × + ×
+
+ =
×
( )
( )
( )
cos - cos cos sin sin
sin - sin cos - cos sin
tan - tan
tan -
1 tan tan
a b a b a b
a b a b a b
a b
a b
a b
= × + ×
= × ×
=
+ ×
W
cos 2 cos² - sin²
2cos² - 1
1 - 2sin²
sin 2 2sin cos
2 tan
tan 2
1 - tan²
a a a
a
a
a a a
a
a
a
=
=
=
= ×
=
1 cos 2
cos²
2
1 - cos 2
sin²
2
a
a
a
a
+
=
=
Wtan
2
a
t =
2
sin
1 ²
1 - ²
cos
1 ²
2
tan
1 - ²
t
a
t
t
a
t
t
a
t
=
+
=
+
=
ġ ħ ĜWħ ġ ĜW
( ) ( )[ ]
( ) ( )[ ]
( ) ( )[ ]
( ) ( )[ ]
1
cos cos cos cos -
2
1
sin sin cos cos
2
1
sin cos sin sin
2
1
cos sin sin - sin
2
a b a b a b
a b a b a b
a b a b a b
a b a b a b
× = + +
× = − + − −
× = + − −
× = + −
cos cos 2cos cos
2 2
cos cos 2sin sin
2 2
sin sin 2sin cos
2 2
sin sin 2cos sin
2 2
p q p q
p q
p q p q
p q
p q p q
p q
p q p q
p q
+ −     + =        
+ −     − = −        
+ −     + =        
+ −     − =        
ĜWcos sina x b x+( ) ( ), 0,0a b ≠
( )
cos sin ² ² cos sin
² ² ² ²
² ² cos
a b
a x b x a b x x
a b a b
a b x α
 + = + +   + +
= + −
αĢW
sin
² ²
b
a b
α =
+
cos
² ²
a
a b
α =
+